Sugarcane juice can be got (freshly squeezed in front of your face) at a cart in front of Mitsuwa Marketplace in Costa Mesa, or else sometimes at a cart in front of Hong Kong Superstore in San Gabriel. I'm sure there are more places but these are the two that I know of.

"Fresh" dates (they're still wrinkly) can be got from the date vendor at the Saturday morning market across from UCI (behind In-N-Out at Campus and Bridge Sts.)

kind of depends on what you mean by "fresh". not to get all pedantic, but the dates in the market right now are dried. truly fresh dates are available only for a fairly short period in August-September.